Validation data are typically preferred, but are often unavailable or impossible to collect. There is literature which uses instruments for the correction of measurement error effects, some of which extends the idea of regression calibration [1]}, [2]}. However, the conditions required for a variable to be viewed as an instrumental variable are often difficult to verify in practice, and falsely assuming that a factor is an instrument can lead to large biases, even when the bias introduced from the measurement error itself is small [2]}. Consequently, most of the literature has focused on the use of replicate measurements. When using replicate data, we assume that we have a sample in which we observe \(\lbrace Y_i, Z_i, X_{i1}^*,\hdots ,X_{i\kappa _i}^*\rbrace \) , where \(\lbrace X_{i1}^*,\hdots ,X_{i\kappa _i}^*\rbrace \)  are independent and identically distributed.
